Calculation Of Natural Frequencies Research Articles

Calculation of natural frequencies of doubly curved laminated shells using a modified higher order zigzag theory

In this study, the analysis of free vibrations for doubly curved laminated shells has been performed with modified higher order zigzag theory (MHOZT). This approach incorporates higher order displacement kinematics and considers both transverse normal and shear strains. The model assumes that the displacement fields within the plane are a combination of a global cubicly varying field and a locally zigzag linearly varying field. On the other hand, the out-of-plane displacement field varies quadratically with shell thickness coordinates. This formulation considers extended thickness criteria, ensuring the inclusion of the ratio of thickness to the radius of curvature (z/R) in strain displacement relations. Additionally, this model guarantees that there is no transverse shear stress at the extreme surfaces of the shell and that inter-laminar shear stress at interfaces is continuous. To implement this formulation effectively, a C° finite element is used. The outcomes are compared against a three-dimensional (3D) elasticity solution, as well as other pertinent results available in the literature. The suggested model accurately predicts the vibration characteristics of laminated shells, showing good agreement with the 3D elasticity solutions.

Improved FEM Natural Frequency Calculation for Structural Frames by Local Correction Procedure

The accurate calculation of natural frequencies is important for vibration and earthquake analyses of structural frames. For this purpose, it is necessary to discretize each beam or column of the frame into one or more smaller elements. The required number of elements per member increases when the frame’s modal shapes have wavelengths similar to the beam lengths. This paper presents a method that reduces the number of elements needed for a precise calculation. This is achieved by implementing a straightforward local correction to the kinetic and elastic energy of certain elements, resulting in a substantial decrease in error. The validity of this method is demonstrated through a range of examples, from simple canonical cases to more realistic ones. Additionally, the paper discusses the unique features of this method and examines its relationship with other approaches.

Suitability of Polymers for 3D-Printing Laboratory Models for Shaking Table Experiments: Discussion and Indications.

This paper presents a comprehensive assessment of the suitability of seven commercially available polymers for crafting laboratory models designed for dynamic shaking-table tests using 3D-printing technology. The objective was to determine whether 3D-printed polymer models are effective for dynamic assessments of structures. The polymers underwent experimental investigations to assess their material properties, i.e., the elastic modulus, the mass density, and the limit of linear-elastic behaviour. The following methodology was applied to obtain the correct values of elasticity moduli and yield points of the polymers: (1) the uniaxial tensile test, (2) the compression test, and (3) the three-point loading test. The filament density was determined as the ratio of sample mass to its volume. The results indicate substantial variations in stiffness, density, and elasticity limits among them. For the similarity analysis, an existing reinforced concrete chimney 120 m high was chosen as a prototype. A geometric similarity scale of 1:120 for a laboratory mock-up was adopted, and a numerical model of the mock-up was created. The similarity scales were calculated for mock-ups made of each filament. Based on these scales, numerical calculations of natural frequencies and dynamic performance under a strong earthquake were carried out for models made of different polymers. Assessment of the polymers' suitability for laboratory models revealed positive outcomes. The agreement between field experiments, shaking-table tests, and numerical predictions in terms of natural frequencies was observed. Maximum stresses resulting from the earthquake indicated the satisfactory performance of the model below the linear-elastic limit. Despite differences in material properties, the selected polymers were deemed suitable for 3D-printing models for shaking-table tests. However, the discussion raised some important considerations. The upper frequency limit of the shaking-table imposes restrictions on the number of natural frequencies that can be determined. Numerical assessments of natural frequencies are recommended to prevent underestimation and to assess the feasibility of their determination. Additionally, resonance during natural frequency determination may lead to exceeding the linear-elastic limit, affecting filament properties, and making the similarity criteria invalid. Practically, this research contributes insights for planning shaking-table tests, aiding in selecting the most suitable filament and highlighting crucial considerations to ensure reliable and accurate dynamic assessments.

Experimental study on fundamental frequency and human‐induced vibration characteristics of light steel foam concrete composite floor

SummaryIn order to study the dynamic characteristics and human‐induced vibration response of light steel foamed concrete composite floor (LCSF), the vibration characteristics of 4.2 m × 5 m LCSF model were tested under the condition of opposite side support, and the natural vibration frequency of LCSF was obtained. The orthogonal anisotropic elastic plate and the simulated beam element were used, respectively, to compute the natural vibration frequency of the LCSF, and the estimated findings were compared with those obtained from measurements. The discrepancy between the calculated results and the measured results of the approximate beam element using the natural vibration frequency calculation technique is around 13%, but the error for the plate element using the calculated results and the tested results is about 23%. To examine the floor's vibration response under the factors of step frequency, walking path, pedestrian density, and load distribution, the LCSF underwent a human‐induced vibration test. The test results show that the fundamental frequency of the LCSF specimen is about 11 Hz, which can meet the requirements of the specification. However, the vibration response of the LCSF specimen under different conditions of pedestrian load is significantly different. With the acceleration of step frequency and the increase and concentration of load, the floor's vibration response becomes more visible. In the route test, it is established that LCSF has the attribute of a unidirectional plate. The natural vibration frequency of similar floor slab can be calculated by the method of simulating the natural vibration frequency of beam element. The findings can serve as a guide for LCSF research and implementation.

Asymptotic long-wave model for a high-contrast two-layered elastic plate

The paper is concerned with the derivation of asymptotically consistent equations governing the long-wave flexural response of a two-layered rectangular plate with high-contrast elastic properties. In the general case, the plate is under dynamic and variable surface, volume, and edge forces. Performing the asymptotic integration of the three-dimensional (3D) elasticity equations in the transverse direction and satisfying boundary conditions on the faces and interface, we derived the sequence of two-dimensional (2D) differential equations with respect to required functions in the first two approximations. The eight independent restraints for the generalized displacements and stress resultants are considered to formulate the 16 independent variants of boundary conditions. One of the main results of the paper is the Timoshenko–Reissner type equation capturing the effect of the softer layer and taking into account the in-plane deformation induced by the edge forces. Comparative calculations of natural frequencies were carried out based on our and alternative models.

Analysis of the dynamic behaviour of lightweight floor slabs

Floor slabs are one of the integral load-bearing elements in buildings. When it comes to their serviceability limit state (SLS), one of the key aspects is their dynamic behaviour. In case of residential or administrative buildings, the knowledge of dynamic behaviour of slabs is important in terms of hygiene (psyche) of the user. When speaking of buildings such as medical facilities or laboratories that are using delicate machinery, the knowledge of the dynamic behaviour is important in order to prevent vibration induced damage to given machinery. Therefore, the aim of this paper is to provide a brief overview of standards applicable for assessment of dynamic behaviour, present various methods for calculation of first natural frequency, as well as to present outlines for in-situ measurements, all with special regard to voided slabs.

Використання моделі балістичного маятника в аналізі перехідних процесів кранових систем

A number of factors affecting the accuracy of experimental determination of small velocities of pendulum vibrations of the load on the rope of the crane hoisting mechanism, arising at start-up, braking or reversing of the crane system, have been analyzed. The rope is considered as an elastic thread. The influence of finite mass and bending elasticity of the viscus on the frequency of natural oscillations of the pendulum is theoretically analyzed. It is shown that the main factor influencing the natural frequency is the bending elasticity of the viscus thread, and the influence of the mass of the thread (rope) of the ballistic pendulum can be minimized. An exact analytical expression for the natural frequency of a real pendulum at arbitrary stiffness of the lightweight viscus is obtained. For small mass of cargo and speed of movement at impulse influence on the crane system (starting, braking, reversing) there are certain difficulties in taking into account the final mass and stiffness of the viscus for calculation of natural frequency of pendulum oscillations. In an ideal situation, the mass of the temple can be made too small compared to the mass of the pendulum (load and rope), but with a short length of rope increases the natural frequency of the pendulum, which reduces the sensitivity of the system to external impulsive influences, and increases the influence of the elasticity of the thread (rope) on this frequency. Therefore, in a real situation, the mass of the pendulum, its length (of the rope) and the stiffness of the temple should always be chosen and taken into account in accurate studies (and calculations) based on compromise considerations. In this study, the degree of influence of the above factors on the natural frequency of oscillation of a pendulum (i.e., essentially a weight on a rope) is theoretically evaluated in the small-parameter approximation. The results obtained in this study can be further used to refine and improve the existing engineering methods of calculation and analysis of transients (start-up, braking, reversing, etc.) of crane systems both at the stages of their design and in the modes of real operation to optimize the performance of the above systems (and mechanisms).

Advanced Frequency Study of Thick FGM Cylindrical Shells by Using TSDT and Nonlinear Shear

For the advanced frequency study of thick functionally graded material (FGM) circular cylindrical shells, it is interesting to consider the extra effects of nonlinear coefficient term in third-order shear deformation theory (TSDT) of displacements on the calculation of varied shear correction coefficient. The formulation for the advanced nonlinear shear correction coefficient are based on the energy equivalence principle. The values of nonlinear shear correction coefficient are usually functions of nonlinear coefficient term of TSDT, power-law exponent parameter and environment temperature. The free vibration frequencies of thick FGM circular cylindrical shells are investigated with the simply homogeneous equation by considering that simultaneous effects of the TSDT, the nonlinear shear correction coefficient of transverse shear force and the two direction of mode shapes. The novelty is more important and reasonable for the thick FGM circular cylindrical shells, especially for the ratio of length to thickness is five by considering the effects of advanced nonlinear shear correction coefficient and nonlinear coefficient term of TSDT on the advanced calculation of fundamental first natural frequencies.

Natural frequencies of prismatic and tapered composite girders with corrugated steel webs

The present study proposed an analytical formula to predict the natural frequencies of prismatic and tapered composite girders with corrugated steel webs (CSWs), and rigorously provides the derivation process. It is experimentally and numerically verified that the proposed formula is more applicable for predicting the natural frequencies of girders with CSWs, in comparison to the traditional formula based on Euler-Bernoulli beam theory. The study demonstrates that the traditional formula used for determining the natural frequencies of prismatic girders with CSWs leads to significant overestimation, mainly because it ignores the effects of shear deformation. The finite element (FE) simulation suggests that the free vibration mode shapes of tapered girders with CSWs exhibit sinusoidal patterns. Consequently, an equivalent stiffness method has been developed for accurately and conveniently estimating the natural frequencies of tapered girders with CSWs. This paper establishes a theoretical foundation for modifying the natural frequency calculation formula of prismatic and tapered girders with CSWs in current specifications, the research findings hold significant implications for the structural dynamic analysis and design of such girders.

A magnetic field model of rectangular permanent magnets in a magnetic pendulum array

The magnetic pendulum array (MPA) is a novel mechanical antenna for efficient and portable ultra-low-frequency transmission. Its natural frequency is affected by the magnetic field of its rectangular permanent magnets. In this work, the influence of the magnetization of a permanent magnet on the demagnetizing field and the magnetic field of other permanent magnets is analyzed to accurately calculate the natural frequency, and a magnetic field model of the rectangular permanent magnets in an MPA is established to improve the calculation accuracy. The consistency of the theoretical model results is verified through experiments and simulations. Finally, the effect of sizes and distances of the permanent magnets on the natural frequency of an MPA is analyzed, providing theoretical guidance for the dynamic analysis of “A Mechanically Based Antenna (AMEBA).” This research has significance and practical application value in the optimization of the magnetic field characteristics and natural frequency calculation methods for AMEBA.

METHODS FOR CALCULATING FREE OSCILLATIONS OF BEAM SYSTEMS

Some methods for calculating free oscillations of mechanical systems with distributed parameters are described with the implementation of these methods on a concrete example. Systems may include point masses and springs. Methods of direct integration of the free oscillation equation, the method of initial parameters and the finite element method are considered. Recommendations to algorithm and calculations of natural frequencies and modes are given. For the finite element method, the influence of the number of elements on the accuracy of determining the spectrum of natural oscillations is investigated.

Consideration of supports stiffness at torsional and longitudinal vibrations of the rod

The study of the impact of the stiffness of the supports on the value of the first natural frequency of torsional and longitudinal free vibrations of the spring-hinged rod was carried out. In both cases of vibrations the mathematical formulation of the problem of determining the first natural frequency of vibrations reduces to compiling and solving the corresponding differential equation under given boundary conditions. The solution of this complex mathematical problem is performed by the numerical method in a specially created program for many combinations of supports stiffness. To obtain an analytical solution, a quadratic approximation of the obtained values was performed, which made it possible to obtain simple dependencies convenient for engineering applied calculations. The quadratic approximation also allowed solving the inverse problem of determining the required stiffness of supports for the required value of the first natural frequency of vibrations. The obtained solutions were compared with known solutions, which showed that the approximation error is less than 5% if the stiffness of the rod supports of the same order. The obtained results can be used in engineering applied calculations of natural frequencies of torsional and longitudinal vibrations of rods with elastic supports.

Simplified Calculation Method for the Natural Frequencies of the Offshore Wind Turbine Structures with Monopile Foundations Based on Euler–Bernoulli Beam Theory

The wind and wave loads on the offshore wind-turbine (OWT) structures with monopile foundations occur at specific frequencies. When these excitation frequencies are close to the natural frequencies of OWT structures, resonance can disturb the proper operation of the power-generation equipment and shorten the service life of the structural system. Therefore, to ensure safe and efficient operation, the natural frequencies of the OWT structures with monopile foundations must be determined. To this end, a calculation method for the natural frequencies of the OWT structures with monopile foundations is proposed. This method, which considers the mass ratio (the ratio of the lumped mass on the top of the tower to the total mass of the tower, the transition piece, and the monopile above the mudline) as well as the nonuniform moment of inertia of the structures above the mudline and the soil–structure interaction (SSI), is derived using the Euler–Bernoulli beam differential equation and slope-displacement equation of a monopile according to the flexibility matrix, stiffness equivalent principle, and virtual work principle. Finally, the natural frequency calculation method is compared with finite element simulation and other calculation methods.

Study on the natural frequency of box girders with corrugated steel webs

In this study, an analytical formula for predicting the natural frequencies of box girders with corrugated steel webs (CSWs) was proposed based on D'Alembert's principle and Timoshenko beam theory. The proposed formula was validated through experiments conducted by previous researchers and FE simulations. The results demonstrated that the proposed formula is more applicable for predicting the natural frequencies of box girders with CSWs compared with the traditional natural frequency calculation formula based on Euler-Bernoulli beam theory. The study found that the traditional formula, ignoring shear deformation and moment of inertia, has a considerable error in overestimating the natural frequencies of box girders with CSWs. A parameter study was performed to investigate the influences of the height to span ratio, the corrugation angle and the thickness of CSWs on the natural frequencies of box girders with CSWs. This paper provides a theoretical basis for the modification of the natural frequency calculation formula of box girders with CSWs in current specifications, research findings can also provide basis for structural dynamic analysis and design.

Creation and research of internal combustion engines dynamic models in OpenModelica

Problem. Torsional vibrations of the internal combustion engine crankshaft are the result of uneven torque due to gas pressure in the cylinders, as well as due to inertial forces of engine moving parts. To analyze torsional vibrations, it is necessary to have an adequate mathematical model of the engine mechanical system that takes its dynamic characteristics into account. Goal. Developing methodology for determining the mass-inertial parameters of connecting rod mechanisms in the internal combustion engines and building dynamic models based on the Openmodelica environment. Developing the Modelica language algorithm for calculating natural frequencies and forms of the dynamic model oscillations. Methodology. The elastic and inertial parameters of the dynamic scheme elements are determined by their energy equivalence to the mechanical system. The construction of a dynamic and mathematical model is implemented using the Modelica programming language. Results. Based on the proposed methodology, dynamic models of straight and V internal combustion engines were built in the Openmodelica environment. Calculation of system oscillation natural frequencies and forms was carried out. The results obtained were compared with data from other sources and analyzed from the physical feasibility point of view. Originality. The proposed methodology for determining the mass-inertial parameters of the connecting rod mechanisms in the internal combustion engines and the way to build dynamic models based on them has been shown. An algorithm has been developed in Modelica language, and natural oscillation frequencies and forms were calculated. Practical value. The developed models can be used as a part of the higher-level dynamic machine models, as well as for determining the parameters of engine crankshaft torsional vibrations.

Analytical Calculation Method of Motor Natural Frequencies Based on Equivalent Model of Three-Layer Cylindrical Shell

Accurate calculation of natural frequencies is significant for accurate prediction and suppression of motor vibration and noise. In order to reduce the errors caused by simplified structure in the analytical method, a new analytical calculation method (ACM) of the natural frequencies of the motor is proposed in this paper. The motor enclosure, stator yoke, stator teeth and windings are equivalent to a three-layer cylindrical shell model. The motor feet are equivalent to the axial ribs on the cylindrical shell. The artificial springs are used to simulate arbitrarily complex boundary conditions between the end covers and the enclosure. The combination of zig-zag theory and first-order shear deformation theory (FSDT) is used to calculate the natural frequencies of the equivalent model of the motor proposed in this paper. According to the zig-zag theory model, the respective displacement function and energy equation are established for each cylindrical shell. The different structural dimensions and material properties between the motor enclosure, stator core and windings are considered. The first-order shear deformation theory is adopted in the displacement model of each layer of cylindrical shell, which considers transverse shear deformation, rotational inertia and orthotropy of materials. Finally, the accuracy of the analytical calculation method proposed in this paper is verified by finite element simulation (FES) and hammering method modal test (HMMT).

Riccati transfer equations for fluid structure interaction in liquid-filled piping systems

In this paper, based on the Riccati transfer matrix method (RTMM), the Riccati fluid structure interaction transfer equations (FSIRTE) are established to improve the numerical stability of the classical fluid structure interaction transfer matrix method (FSITMM). Combined with numerical algorithms for eliminating the singularity points of the Riccati equations, the spare root problem in the calculation process is solved. This method can be used for the natural frequency calculation of liquid-filled piping systems. Compared with finite element method (FEM), it has the characteristics of high calculation efficiency; meanwhile, good numerical stability, compared with FSITMM; and accurate calculation results, compared with method of characteristics (MOC). Numerical simulation results of typical classical examples are given.

Approximate calculation of natural frequencies of oscillations of the diamond-shaped plates of the discrete-continuous inter-resonance vibrating table

Conventional single- and two-mass vibrating machines do not have a sufficient level of energy efficiency. There are vibration machines based on inter-resonance oscillating systems, which make possible the significant reduction in the use of electricity. To implement highly effective inter-resonance operation modes of vibrating machines, the oscillating masses of such a system must precisely calculate values of inertial-rigid parameters, as well as their frequency of oscillations. Since the synthesis of continuous sections into classical discrete models of multi-mass mechanical oscillating systems has recently become widespread, the most optimal option is to use a continuous member as a reactive mass of a vibrating machine. The continuous member combines inertial and stiffness parameters and makes it extremely easy to perform the reactive mass. In particular, rectangular plates are used as continuous members. The rectangular shape of the plate is not the only option for making a continuous member. This paper proposes the construction of a discrete-continuous inter-resonance vibration machine with electromagnets, where a diamond-shaped plate is used as a continuous member. The authors carried out the computation of the first natural frequency of the diamond-shaped plate using the Rayleigh-Ritz method with the equation of the hyperboloid. The determined eigenfrequency was confirmed in the Ansys software.

An Explanation for a Paradox in a Fluid-Discharging Cantilevered Pipe Attached With an End-Mass

Abstract For a fluid-discharging cantilevered pipe attached with an end-mass, there are two methods to account for the end-mass effect. The first is that the end-mass is considered in the boundary conditions. The second is that the end-mass is included in the equation of motion via a Dirac delta function. As the analytical solution of the linear free vibration model is not available due to the presence of Coriolis force, the eigenfunctions of a beam, which satisfy the same boundary conditions, are commonly employed in the Galerkin method. It has been found the first method is incorrect for natural frequency calculation when the internal flow velocity is nonzero. However, the intrinsic mechanism remains to be clarified. This study has demonstrated the eigenfunctions in the first method depend on the end-mass and the orthogonality relations are quite different from that of typical simple beams, based on which a new model is proposed and the prediction compares well with that in the second method. For further validation, the critical internal flow velocity, the onset flutter frequency, and the dynamic responses of suspended pipes under gravity are computed, which compare well with experimental observations. This study can provide a workbench for fluid-conveying pipes with various boundary conditions.

Calculation of Stator Natural Frequencies of Electrical Machines Considering Complex Boundary Conditions

The accurate calculation of the natural frequencies of the stator is an important prerequisite for predicting and improving electromagnetic vibration in electrical machines. In this article, the stator core and the enclosure are regarded as cylindrical shells with axial ribs attached inside and outside the shells, respectively. Considering the moment of inertia and the shear deformation of the cylindrical shell, the displacement functions of the mid-surface in cylindrical shell are expressed as a superposition of orthogonal polynomials, the natural frequencies including axial, and circumferential modal shapes under different boundary conditions are obtained by combining the energy method and the Gram–Schmidt orthogonalization method, and the accuracy of the calculation results is verified by the finite element method and modal test. Considering the orthogonality of the stator material and specific slot structure, the natural frequencies of the stator including the stator core, windings, and enclosure are accurately obtained, the actual constraints of the end covers on the enclosure are accurately simulated by the simply supported boundary conditions.